Strange attractors

A couple of months ago, I read Paul J. Steinhardt's wonderful book The Second Kind of Impossible, about his (and others') search for quasicrystals -- a bizarre form of matter that is crystalline but aperiodic (meaning it fills the entire space in a regular fashion, but doesn't have translational symmetry).  Here's an artificial quasicrystal made of aluminum, palladium, and manganese:

[Image is in the Public Domain courtesy of the United States Department of Energy]

As the above photograph shows, they can be created in the lab, but Steinhardt believed they could occur naturally -- and he finally proved it, in a meteorite sample he and his team found in a remote region of Siberia.

I was immediately reminded of Steinhardt's aperiodic crystals when I read a paper in Chaos: An Interdisciplinary Journal of Nonlinear Science, by Francesca Bertacchini, Pietro Pantano, and Eleanora Bilotta, of the University of Calabria, who were experimenting with another nonrandom but chaotic shape -- a "strange attractor."

A strange attractor is a concept from fractals and chaos theory, and represents a value toward which a perturbed system tends to evolve.  Chaos theory has been around for a while, but came to most people's attention from Jurassic Park, when the character Ian Malcolm (portrayed in memorable fashion by Jeff Goldblum) is explaining the unpredictability of complex systems using the direction a drop of water rolls on a relatively (but not perfectly) flat surface, in this case, the back of someone's hand.  Systems like that one tend to rush far out of equilibrium -- once the drop starts to move, it keeps going -- but some systems settle into a set of loops or spirals, as if something in the middle was drawing them in.

Thus the name strange attractor.

These systems, when mapped out, create some beautiful patterns -- like Steinhardt's quasicrystals, with the superficial appearance of regularity, but without any repeats or obvious symmetries.  Bertacchini et al. used the mathematical functions describing the system to drive a 3-D printer and actually create models of what strange attractors look like.  The team was struck with how beautiful the shapes were, and had a goldsmith fashion them as jewelry.  Here are a few of their creations:

They look a little like Spirograph patterns gone off the rails, but they have a striking, almost-but-not-quite-symmetrical shape that keeps drawing the eye back.

The authors write:

[We used] a chaotic design approach used to develop jewels from chaotic design.  After presenting some of the most important physical systems that generate chaotic attractors, we introduced the basic steps of this approach.  This approach exploits a number of fundamental characteristics of chaotic systems.  In particular, the parametric design approach exploits the concept of extreme sensitivity to the initial data that leads to evolutionary transformations of dynamic systems, not only along the traditional routes to chaos and through qualitative changes in the starting chaotic system, but also through changes in the basic parameters of the system, which create infinite chaotic forms.  Such phase spaces, therefore, represent an enormous potential to be exploited in the design of artistic objects, whether they are jewelry pieces or other objects of abstract art. In the computational approach used, each shape is unique and it is identified by a set of parameters that almost constitute its precise value.  This leads to the creation of unique artistic forms and, thus, to the customization of products in the case of jewelry pieces, which exploits chaotic design as a methodology.

The whole thing brings up for me the mysterious question of what we find beautiful -- and how so often, it's a balance between predictability and unpredictability, between symmetry and randomness.  It reminds me of the quote from the brilliant electronic music pioneer Wendy Carlos: "What is full of redundancy is predictable and boring.  What is free from all structure is random and boring.  In between lies art."

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Descent into chaos

There's an interesting concept called sensitive dependence on initial conditions.

Here's a simple example.  If you take a deep bowl, and drop a marble into it, it doesn't take any great intelligence or insight to predict what the end state will be.  Marble on the bottom of the bowl.  It doesn't matter how high you drop it from or where exactly it hits the sides first.  After a bit of rolling around, the marble will stop moving at the bottom.

Now, do the same thing -- but with the bowl flipped over.  Where will the marble end up?

Impossible to say, because it is an inherently chaotic system.  You could do it a hundred times and the marble will end up in a different place each time, because its final location depends on exactly the speed and angle of its path, where it hits the curved edge of the bowl, even whether the marble is spinning a little or not.  A system like this is said to be "sensitive to initial conditions" -- therefore unpredictable.  Perturb it a little by altering it in a tiny way, and you get a completely different outcome.

Here's a much cooler example, that I stumbled across in doing research for this post.  It's called a double compound pendulum.  Take two rigid rods, and suspend one so it's free to swing.  Then tie the second rod to the bottom of the first.  Start with the rods pulled horizontal, then let it go.  Can you predict how the whole system will move?

Simple answer: no.  It's a chaotic system.

[GIF is in the Public Domain]

A little mesmerizing to watch, isn't it?

The reason this comes up is because there's decent evidence that the intersection between the Earth's climate and human society is a chaotic system that has at least some degree of sensitive dependence to initial conditions.  If you perturb it, it may not respond the way you expect -- and sometimes small changes in one location can lead to big ones somewhere else.  (This concept was made famous as "the butterfly effect.")

As an example of this, take the research that was released just last week in Proceedings of the National Academy of Sciences, the link to which was sent to me by a friend and loyal reader of Skeptophilia yesterday.  In "Extreme Climate After Massive Eruption of Alaska’s Okmok Volcano in 43 BCE and Effects on the Late Roman Republic and Ptolemaic Kingdom," by a team led by Joseph R. McConnell of the University of Cambridge, we find out about an Alaskan volcanic eruption that may have been one of the significant factors leading to the collapse of the Roman Republic, and its consolidation as an empire -- events that radically changed the course of history in Europe and North Africa.

Geologists on the team identified tephra (volcanic ash) in ice cores from the Arctic that were fingerprinted chemically and shown to come from the volcano named Okmok in the Aleutian Islands.  The dating of the tephra deposit shows that the eruption happened in 43 B.C.E. -- right after the assassination of Julius Caesar, during a time when Rome was in chaos as various political factions were duking it out for control.  The eruption of this volcano halfway around the world is also correlated with the coldest year Europe had for centuries, possibly longer.  Snow fell in summer, crops failed, there were famines and repeated uprisings by desperate and starving citizens.

This sudden drop in temperature was one of the factors that contributed to the realignment of the Roman government as someone emerged who said he knew what to do to fix the situation -- Octavian (later known as Augustus), Julius Caesar's great-nephew.  And he did it, establishing the Pax Romana, quelling the revolts and ushering in two centuries of relative peace and prosperity for Roman citizens (and wreaking havoc on the Gauls, Celts, Teutons, and whatever other tribes happened to be in the way of the Roman Legions).

It helped, of course, that once the volcanic tephra from Okmok settled out, the temperature rebounded, and the first years of Augustus's reign were noted for a beneficent climate and rich crop yields.  Not all of the good bits of the Pax Romana were due to Augustus's skill as an emperor; he got lucky because of conditions he had no control over and could not have predicted, just as the last leaders of the Republic got unlucky for the same reasons.

The point here is that we should be wary of perturbing chaotic systems, which is exactly what we're doing by our rampant dumping of carbon dioxide into the atmosphere.  And what we're seeing over the last decades is exactly the sort of unpredictable response -- some areas experiencing droughts, others floods; deadly heat waves and trapped polar vortexes that drop areas into the deep freeze for weeks; increased hurricanes, tornadoes, and bomb cyclones.  One of the frustrations felt by the people who understand climate systems is that the average layperson doesn't see this kind of  unpredictability as precisely what you'd expect from pushing on an inherently chaotic system.  If you can't make predictions to pinpoint accuracy -- "okay, because the climate is changing, you can expect it to be 95 F in Omaha on July 19" -- it's nothing to be concerned about.

"The scientists don't even know what's going on," you'll hear them say.  "Why should we believe it's a problem if they can't tell us what the outcome is going to be?"

But that's exactly why we shouldn't be messing with it.  Systems that have sensitive dependence to initial conditions are dramatically unpredictable, and get pushed out of equilibrium quickly and sometimes with catastrophic results.

As the leaders in the final years of the Roman Republic found out.

I feel like another figure from the Classical world -- Cassandra -- for even bringing this up.  Cassandra, you may recall, is the woman who was cursed by the gods to having accurate foresight and knowledge of the future, but with the difficulty that whatever she says, no one believes.  The climatologists have been sounding the alarm about this for decades, to little effect.  If you can't accurately predict the outcome, to most politicians, it doesn't exist.

Which makes me wonder if before we try to get our leaders to get on board with addressing anthropogenic climate change, we should require they sit through some lectures on chaos theory.

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I know I sometimes wax rhapsodic about books that really are the province only of true science geeks like myself, and fling around phrases like "a must-read" perhaps a little more liberally than I should.  But this week's Skeptophilia book recommendation of the week is really a must-read.

No, I mean it this time.

Kathryn Schulz's book Being Wrong: Adventures in the Margin of Error is something that everyone should read, because it points out the remarkable frailty of the human mind.  As wonderful as it is, we all (as Schulz puts it) "walk around in a comfortable little bubble of feeling like we're absolutely right about everything."  We accept that we're fallible, in a theoretical sense; yeah, we all make mistakes, blah blah blah.  But right now, right here, try to think of one think you might conceivably be wrong about.

Not as easy as it sounds.

She shocks the reader pretty much from the first chapter.  "What does it feel like to be wrong?" she asks.  Most of us would answer that it can be humiliating, horrifying, frightening, funny, revelatory, infuriating.  But she points out that these are actually answers to a different question: "what does it feel like to find out you're wrong?"

Actually, she tells us, being wrong doesn't feel like anything.  It feels exactly like being right.

Reading Schulz's book makes the reader profoundly aware of our own fallibility -- but it is far from a pessimistic book.  Error, Schulz says, is the window to discovery and the source of creativity.  It is only when we deny our capacity for error that the trouble starts -- when someone in power decides that (s)he is infallible.

Then we have big, big problems.

So right now, get this book.  I promise I won't say the same thing next week about some arcane tome describing the feeding habits of sea slugs.  You need to read Being Wrong.

Everyone does.

[Note: if you purchase this book using the image/link below, part of the proceeds goes to support Skeptophilia!]